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luke naylor latex documents
research
Max Destabilizer Rank
Commits
7bd81231
Commit
7bd81231
authored
1 year ago
by
Luke Naylor
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Add plot for characteristic curves in rank 0 case
parent
7d7ebf5b
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main.tex
+61
-21
61 additions, 21 deletions
main.tex
rank_zero_case.ipynb
+23
-15
23 additions, 15 deletions
rank_zero_case.ipynb
with
84 additions
and
36 deletions
main.tex
+
61
−
21
View file @
7bd81231
...
...
@@ -282,9 +282,7 @@ Considering the stability conditions with two parameters $\alpha, \beta$ on
Picard rank 1 surfaces.
We can draw 2 characteristic curves for any given Chern character
$
v
$
with
$
\Delta
(
v
)
\geq
0
$
and positive rank.
These are given by the equations
$
\chern
_
i
^{
\alpha
,
\beta
}
(
v
)=
0
$
for
$
i
=
1
,
2
$
, and
are illustrated in Fig
\ref
{
fig:charact
_
curves
_
vis
}
(dotted line for
$
i
=
1
$
, solid for
$
i
=
2
$
).
These are given by the equations
$
\chern
_
i
^{
\alpha
,
\beta
}
(
v
)=
0
$
for
$
i
=
1
,
2
$
.
\begin{definition}
[Characteristic Curves
$
V
_
v
$
and
$
\Theta
_
v
$
]
Given a Chern character
$
v
$
, with positive rank and
$
\Delta
(
v
)
\geq
0
$
, we
...
...
@@ -296,9 +294,21 @@ define two characteristic curves on the $(\alpha, \beta)$-plane:
\end{align*}
\end{definition}
\begin{fact}
[Geometry of Characteristic Curves]
\subsection
{
Geometry of the Characteristic Curves
}
These characteristic curves for a Chern character
$
v
$
with
$
\Delta
(
v
)
\geq
0
$
are
not affected by flipping the sign of
$
v
$
so it's only necessary to consider
non-negative rank.
As discussed in subsection
\ref
{
subsect:relevance-of-V
_
v
}
, making this choice
has Gieseker stable coherent sheaves appearing in the heart of the stability
condition
$
\firsttilt
{
\beta
}$
as we move `left' (decreasing
$
\beta
$
).
\subsubsection
{
Positive Rank Case
}
\label
{
subsect:positive-rank-case-charact-curves
}
\begin{fact}
[Geometry of Characteristic Curves in Positive Rank Case]
The following facts can be deduced from the formulae for
$
\chern
_
i
^{
\alpha
,
\beta
}
(
v
)
$
as well as the restrictions on
$
v
$
:
as well as the restrictions on
$
v
$
, when
$
\chern
_
0
(
v
)
>
0
$
:
\begin{itemize}
\item
$
V
_
v
$
is a vertical line at
$
\beta
=
\mu
(
v
)
$
\item
$
\Theta
_
v
$
is a hyperbola with assymptotes angled at
$
\pm
45
^
\circ
$
...
...
@@ -312,21 +322,8 @@ as well as the restrictions on $v$:
\end{itemize}
\end{fact}
\subsection
{
Relevance of the Vertical Line
$
V
_
v
$}
By definition of the first tilt
$
\firsttilt\beta
$
, objects of Chern character
$
v
$
can only be in
$
\firsttilt\beta
$
on the left of
$
V
_
v
$
, where
$
\chern
_
1
^{
\alpha
,
\beta
}
(
v
)
>
0
$
, and objects of Chern character
$
-
v
$
can only be
in
$
\firsttilt\beta
$
on the right, where
$
\chern
_
1
^{
\alpha
,
\beta
}
(-
v
)
>
0
$
. In
fact, if there is a Gieseker semistable coherent sheaf
$
E
$
of Chern character
$
v
$
, then
$
E
\in
\firsttilt\beta
$
if and only if
$
\beta
<
\mu
(
E
)
$
(left of the
$
V
_
v
$
), and
$
E
[
1
]
\in
\firsttilt\beta
$
if and only if
$
\beta\geq\mu
(
E
)
$
.
Because of this, when using these characteristic curves, only positive ranks are
considered, as negative rank objects are implicitly considered on the right hand
side of
$
V
_
v
$
.
These are illustrated in Fig
\ref
{
fig:charact
_
curves
_
vis
}
(dotted line for
$
i
=
1
$
, solid for
$
i
=
2
$
).
\begin{sagesilent}
from characteristic
_
curves import
\
...
...
@@ -334,6 +331,7 @@ typical_characteristic_curves, \
degenerate
_
characteristic
_
curves
\end{sagesilent}
\begin{figure}
\centering
\begin{subfigure}
{
.49
\textwidth
}
...
...
@@ -358,8 +356,49 @@ degenerate_characteristic_curves
\label
{
fig:charact
_
curves
_
vis
}
\end{figure}
\begin{sagesilent}
from rank
_
zero
_
case import Theta
_
v
_
plot
\end{sagesilent}
\subsubsection
{
Rank Zero Case
}
\begin{fact}
[Geometry of Characteristic Curves in Rank 0 Case]
The following facts can be deduced from the formulae for
$
\chern
_
i
^{
\alpha
,
\beta
}
(
v
)
$
as well as the restrictions on
$
v
$
, when
$
\chern
_
0
(
v
)=
0
$
and
$
\chern
_
1
(
v
)
>
0
$
:
\begin{minipage}
{
0.49
\textwidth
}
\begin{itemize}
\item
$
V
_
v
=
\emptyset
$
\item
$
\Theta
_
v
$
is a vertical line at
$
\beta
=
\frac
{
D
}{
C
}$
where
$
v
=
\left
(
0
,C
\ell
,D
\ell
^
2
\right
)
$
\end{itemize}
\end{minipage}
\begin{minipage}
{
0.49
\textwidth
}
\sageplot
[width=\textwidth]
{
Theta
_
v
_
plot
}
%\caption{$\Delta(v)>0$}
%\label{fig:charact_curves_rank0}
\end{minipage}
\end{fact}
\subsection
{
Relevance of the Hyperbola
$
\Theta
_
v
$}
\subsection
{
Relevance of
$
V
_
v
$}
\label
{
subsect:relevance-of-V
_
v
}
By definition of the first tilt
$
\firsttilt\beta
$
, objects of Chern character
$
v
$
can only be in
$
\firsttilt\beta
$
on the left of
$
V
_
v
$
, where
$
\chern
_
1
^{
\alpha
,
\beta
}
(
v
)
>
0
$
, and objects of Chern character
$
-
v
$
can only be
in
$
\firsttilt\beta
$
on the right, where
$
\chern
_
1
^{
\alpha
,
\beta
}
(-
v
)
>
0
$
. In
fact, if there is a Gieseker semistable coherent sheaf
$
E
$
of Chern character
$
v
$
, then
$
E
\in
\firsttilt\beta
$
if and only if
$
\beta
<
\mu
(
E
)
$
(left of the
$
V
_
v
$
), and
$
E
[
1
]
\in
\firsttilt\beta
$
if and only if
$
\beta\geq\mu
(
E
)
$
.
Because of this, when using these characteristic curves, only positive ranks are
considered, as negative rank objects are implicitly considered on the right hand
side of
$
V
_
v
$
.
\subsection
{
Relevance of
$
\Theta
_
v
$}
Since
$
\chern
_
2
^{
\alpha
,
\beta
}$
is the numerator of the tilt slope
$
\nu
_{
\alpha
,
\beta
}$
. The curve
$
\Theta
_
v
$
, where this is 0, firstly divides the
...
...
@@ -377,6 +416,7 @@ $\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on
$
\Theta
_
v
$
, and hence the apex of the circular pseudo-wall with centre
$
(
\beta
,
0
)
$
(as per subsection
\ref
{
subsect:bertrams-nested-walls
}
).
\subsection
{
Bertram's Nested Wall Theorem
}
\label
{
subsect:bertrams-nested-walls
}
...
...
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